Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T06:16:06.138Z Has data issue: false hasContentIssue false
  • English
  • Français

ELEMENTS OF HIGH ORDER ON FINITE FIELDS FROM ELLIPTIC CURVES

Part of: Arithmetic problems. Diophantine geometry Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  02 March 2010

JOSÉ FELIPE VOLOCH
JOSÉ FELIPE VOLOCH*
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

Keywords

finite fieldselliptic curvesmultiplicative group

MSC classification

Secondary: 14G15: Finite ground fields 11T06: Polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 425 - 429
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Burkhart, J. F., Calkin, N. J., Gao, S., Hyde-Volpe, J. C., James, K., Maharaj, H., Manber, S., Ruiz, J. and Smith, E., ‘Finite field elements of high order arising from modular curves’, Des. Codes Cryptogr. 51(3) (2009), 301–314.CrossRefGoogle Scholar
[2]Cohen, S. D., ‘The explicit construction of irreducible polynomials over finite fields’, Des. Codes Cryptogr. 2(2) (1992), 169–174.CrossRefGoogle Scholar
[3]Gao, S., ‘Elements of provable high orders in finite fields’, Proc. Amer. Math. Soc. 127 (1999), 1615–1623.CrossRefGoogle Scholar
[4]Voloch, J. F., ‘On the order of points on curves over finite fields’, Integers 7 (2007), A49.Google Scholar
[5]Wiedemann, D., ‘An iterated quadratic extension of GF(2)’, Fibonacci Quart. 26 (1988), 290–295.Google Scholar